Revisiting the Foundations of Cluster Expansion

The cluster expansion (CE) has been a cornerstone of alloy modeling since the mathematical foundations were laid down more than four decades ago. Since that time, many practical innovations have been developed. While the original mathematical underpinnings (designed for infinite lattices and infinite configurations) are well established, fundamental mathematical and numerical questions remain unanswered for practical problems in which the data and prediction spaces are necessarily finite. For example, it is now possible to formally enumerate all structures up to a given size, but an algorithm for generating a full-rank data matrix, covering both the test and prediction space, has not been published. Also, how can a complete but linearly independent set of clusters be efficiently generated? Should assumptions about the negligibility of long-range or high-vertex interactions be revised? How does the choice of site basis functions affect numerical stability? We present (i) an algorithm for generating a full-rank (and not overcomplete) data matrix, (ii) numerical stability tests for different cluster site functions, and (iii) a new paradigm for predicting empirical risk before data is fit to the model.